Note: The original version of "Exploration of the Earth's Magnetosphere" tried to simplify the presentation by leaving out any discussion of electric fields. Still, electric forces and electric fields play a great role in magnetospheric processes, and as details were added, it became increasingly hard to avoid them

To help fill the gap, section #4a on ""The Electric Fluid" was added, followed by optional section #10a which describes the effect of electric fields on particle motion in the magnetic fields of space. In this section and the one following, another missing component is covered--the induced electromagnetic field. This material is harder and therefore labeled "optional," it actually requires more advanced mathematics, and any math is therefore grossly simplified. And yet, the author hopes these sections may help users understand electromagnetic induction. Do they? Comments welcome!

The Relations between Electricity and Magnetism

Until 1820 studies of electricity and magnetism grew independently.

Electric charges could be generated by friction ("static electricity") and propagated as electric currents along conductors such as metal wires. In 1800 Alessandro Volta discovered a way of creating a steady electric current by means of chemical reactions, in a "voltaic cell" typically containing different metals immersed in an acid or base. (Here we follow popular custom of calling such devices "electric batteries" even though, strictly speaking, a battery is a combination of multiple cells, hooked up together to produce higher voltage).

Meanwhile magnetism was seen as the force produced by special materials, especially some iron-rich minerals or "lodestones," or by steel which had been stroked by an iron magnet or by a lodestone.

Then in 1820 Oersted and Ampere unexpectedly found that the two branches intersected--that electric currents could produce magnetism. That opened the door to a whole range of electrical technology, beginning with the telegraph and telephone.

But there exists a second intersection as well, just as essential. Not only can electric currents create magnetism--magnetic fields can also create electric fields and currents. That is the phenomenon of electromagnetic induction, discussed in what follows.
(Induced electric fields need be distinguished from induced magnetic fields, described in #4 of "The Great Magnet, the Earth").

Induced Electric Currents

Joseph Henry, following Oersted and Ampere, introduced the electromagnet. If you wind a coil of insulated wire around an iron bar, drive an electric current through it (using for instance an electric battery), the bar becomes magnetic. Could the process be perhaps reversed? Wrap a coil of wire around a bar magnet, then close its circuit and look for an electric current flow. No, that does not work--no current flows.

(One could of course point out that if the device did work, it would be equivalent to a perpetual motion machine, an unending source of electric current. Actually, two different phenomena are involved here: (1) to drive a current around a metal wire, we must (by Ohm's law) overcome electric resistance and generate heat, which requires a constant energy input, and (2) the current also creates magnetism. If we could use a "superconducting coil" with no resistance--an effect discovered much later, at very low temperatures--the first effect would disappear, and maybe, just maybe, the inverse experiment could work in some fashion.)

Suppose the experiment is repeated with an electromagnet--a bar of iron surrounded by a "primary" wire coil C1, through which a battery drives some current. Again, that bar also threads a "secondary" coil C2, where we wish to generate a new electric current. But as before, if no changes occur, even if a current flows in C1, none is seen in C2.

However, the situation differs if conditions change with time. If we start by winding the coil around a wooden stick, remove the stick and insert the bar magnet in its place, a momentary current will flow while the bar is inserted. It stops when the magnet comes to a rest inside the coil, but when we pull it out again, another momentary current flows in the opposite direction.

Similarly, with the iron bar threading the two coils C1 and C2. If at first no battery is connected anywhere, nothing happens. The moment we connect the battery to C1 and the iron bar becomes magnetized, a momentary current flows in C2, and if we disconnect it and the magnetism ceases, a momentary current again flows in C2 but in the opposite direction. (We assume the bar does not stay magnetized, which depends on the material and the strength of the current in C1.)

Those are electromagnetically induced currents, originally discovered by Faraday in 1831. Unlike the magnetic field caused by currents, these currents depend on the rate of change of the magnetic field.

Magnetic Flux

You would expect the current in the secondary coil C2 ("secondary current") to depend on the strength of the magnet or electromagnet used, on the number of windings, on the speed at which the magnet is pushed in (or the current in C1 is allowed to grow), and all that is true. That dependence involves the so called magnetic flux through the secondary circuit (called by some teachers "the number of magnetic field lines" passing C2).

Formally the magnetic flux F through a closed loop equals its area, times the average magnetic field perpendicular to it, times the "magnetic permeability" μ of the material inside it. Each of these requires some explanation.

The "area of the loop" sounds simple enough, except that in a coil we count separately each turn threaded by magnetic field lines. Thus with 1000 windings, the effective area threaded by the coil is 1000 times the area of a single turn.

The word "average" conceals a lot of math. You should (from here on) know some of the properties of vectors, quantities which (like force) have a direction in space as well as a magnitude. To find the magnetic flux threaded by a wire loop in empty space (or by any closed curve in space!),

Span it with some surface. (The properties of the magnetic field ensure here it makes no difference which surface is chosen.)

Divide that surface into a large number of itsy-bitsy small patches.

Multiply the area of each patch by the part (vector component) of the magnetic field perpendicular to it, and finally,

Add all those products together to get the magnetic flux.

The "average magnetic field" would be the flux divided by the area covered. Both the flux and the average are concepts from calculus ("obtaining an area integral"), and we won't go into details of practical derivations.

And the permeability μ --most people know that putting a suitable magnetic material inside the loop can concentrate the magnetic field lines. For appropriate iron, μ (strictly, the relative permeability) can be as big as 10,000, but for most materials (wood, copper, glass) it is close to 1, and multiplying by 1 has no effect. In vacuum, of course, μ =1 exactly.

The current in the secondary coil is proportional to the rate at which the magnetic flux threading it changes, no matter what causes the change--whether it is motion relative to the source of the magnetic field, or with primary and secondary coils around the same core, change in the magnetic field of the core because the current in the primary coil rises or falls.

"Proportional" skips over many details--obviously, the electric (ohmic) resistance of the coil also controls the current, and if the current is big (with a wire of low resistance), its own magnetic field also modifies the flux.

Background concept: Vector Fields

The above are the basic facts of electromagnetic induction. For actual calculation, a lot more is needed, well beyond the scope of this web exposition, especially if the circuits and fields are not channeled by copper and iron as they are in electric machinery, but are spread out in three-dimensional space. To get the flavor of what this is about, consider that strange word "flux". It suggests the flow of some fluid, and rightly so.

A magnetic field is a region in space where magnetic forces may be observed. The region is 3-dimensional and to specify the magnetic field, one must give at each point in the field the direction and magnitude of the magnetic force B on some imaginary isolated magnetic pole there, one of unit strength. B is a vector--a quantity in 3 dimensions with both direction and magnititude. Such quantities are discussed on the web page "vectors" and some other examples of vector quantities are force, acceleration and velocity. Referring to B as a "vector field" means that a region in space exists, at each point of which a vector B is defined.

A map of magnetic field lines is another way of representing the vector field B--the direction of the line at any point is the direction of B, and the density of lines around it is proportional to the intensity. Magnetic field lines crowd together where the magnetic field is strong, for instance near magnetic poles.

An electric field is a 3-dimensional region of electric forces, and the electric vector E is proportional to the force which might be experienced by some charged particle at that point, e.g. by an electron. Other properties are analogous, and one could define electric field lines too, although few people use them. (We omit here the relationship between E and B reflected in the theory of relativity, important for specifying the fields when seen by moving observers).

However, there exists another vector field, studied even earlier--the flow of a fluid, such as air or water. Imagine for instance a swimming pool filled with water, agitated by (say) swimmers in the pool. At each point the water may flow with some velocity v, and the distribution of v is also a vector field, similar to the magnetic field B (the fact water is almost incompressible brings the properties of v and B even closer). One can distinguish 3 types of flow v.

The water can just slosh around.

It can flow in and out from pipes connected to the pool (flow of air, which is compressible, can in addition be also be compressed or expanded). Or else

It can just swirl around on closed paths, Mathematics has expression for each of those, although (1) and (2) overlap. (Math addicts only: look up "Helmholtz Theorem")

Now imagine placing some closed hoop, not necessarily circular, in the pool. How much water (liters, gallons, cubic feet etc.) pass through the hoop each second? That would be the "flux of water" across the hoop, and to derive it, (1) stretch some surface across it (if the compression of water is negligible, any surface gives the same result), (2) divide it into many small areas, (3) multiply each area by the component of v perpendicular to it, and (4)add together all values. Voilà! You have derived the flux of water across the hoop at the given instant. Magnetic flux is mathematically analogous--as if the streamlines of the water flow were replaced by magnetic field lines.

(Optional) Because the flow of fluids is the oldest vector field studied, mathematical expressions which characterize it were developed, symbols or "operators" standing for related calculus expressions (which won't be reproduced). They include the vorticity ∇×v at each point, showing the local tendency to swirl around it (it is a vector directed along the axis of the swirl), and the divergence ∇⋅v giving the rate at which fluid enters at that point (negative if it drains out), either from a pipe entering there or (say, in air) by the expansion of previously compressed fluid. In the study of magnetic and electric fields in space, such operators are useful tools--but at this level, only the general flavor can be given, not the tedious details.
Questions from Users: *** What is "Magnetic Flux" and what are "Flux Lines" *** Magnetic Flux